I am currently DM'ing a campaign, in which one of the PCs is a Forge Cleric.

Artisan's Blessing lets Forge Clerics conduct a ritual to craft non-magical items worth no more than 100gp. Raw materials of equivalent value must be provided in order to craft the item.

A suit of armour is on the list of example items that a Forge Cleric could create using this power. However, there are some suits of Armour (in the medium and heavy armour categories) that are worth more than 100gp. RAW would suggest that a Forge Cleric is unable to craft these items.

My Forge Cleric player asked me if he could get around this limitation by crafting a more expensive suit of armour over multiple uses of Artisan's Blessing. For example he reasoned he could create a suit of Splint armour (worth 200gp) over two uses of Artisan's Blessing, making half at a time.

Now, as I have said, I am pretty convinced that RAW this is not allowed (though I'm open to being convinced otherwise) but what would be the balance implications of allowing this in my game?

The PC would still need to provide sufficient raw materials to make the suit of armour, and the process would become significantly more time consuming. It's also still only providing him with an item which in many campaigns could be easily purchased from any nearby shop. It therefore doesn't seem unreasonable to allow my player to break RAW and craft more expensive items, piece by piece.

Have I missed something, or should this be OK game balance wise?

I don't see why not.

If you can craft an object, you certainly can craft an incomplete object. Say you want to make a necklace, with a very intricate chain. You can certainly craft half that chain at a point, then another half at another. As a DM, you can adjudicate that joining both halfs requires either another use, or specialized work, but I would just allow the Cleric to join both. He is taking the time and resources to craft these things, day by day, just like you would with Downtime Rules.

It is creative, and as far as I can tell, won't break anything.

In your specific case, as as a real life example, consider the Plate Armor (1800g), made out of many parts. The Cleric should be able to craft one of these with every use/pair of uses. After all of them are complete, you have the plate armor, which together is valued at 1800g.

That being said, you can also consider that if the object being crafted is a single piece that is valued over 100g, then since you cannot craft it by parts, then you cannot use Artisan's Blessing with it.

Would it affect your game much? Probably not, as long as you'd technically allow them to purchase the item from a vendor.

As you mentioned, the cleric will have to provide raw materials worth the value of the final item. This basically means that it's the same as buying it from a shop, with the deciding factor being availability. As such, the only scenario I can see where it might affect (your intended) balance is when you intended for a certain item to not exist in your world.

As long as the cleric knows the recipe, which they likely got hammered into them since they chose their domain, and is willing to spend his limited uses per day over multiple days to create the item, I see no major balance issue, even if you allow them to use the ability to fuse all parts together into the complete item (about that, see below).

You likely can create the parts, but not the item.

From the effect of Artisan's Blessing (emphasis mine):

As part of this ritual, you must lay out metal, which can include coins, with a value equal to the creation. The metal irretrievably coalesces and transforms into the creation at the ritual’s end, magically forming even nonmetal parts of the creation.

Basically, you magically (and completely) melt down the raw materials and reshape and transform them into the desired form. As GM, I would not forbid you from using this ability to create parts of an item, as long as each part is worth less than 100G.

However, you cannot use Artisan's Blessing to then combine the parts into a single item worth more than 100G (since this essentially means magically melting down and reshaping the parts you had made just before. The ability doesn't care what the metal used for the ritual looks like). As such, as a GM I would require you to use conventional means/tools and appropriate skill checks to put together the parts you made with Artisan's Blessing.

TL;DR: Artisan's Blessing for Parts, conventional crafting to put them together.